Yes, and it can be simpler to calculate it, but it tends to have worse theoretical properties. In particular, with log likelihood we can sum the scores for a se

Yes, and it can be simpler to calculate it, but it tends to have worse theoretical properties. In particular, with log likelihood we can sum the scores for a sequence of conditional probability estimates and we get the same score as if it were a single estimate for a single compound random variable. Brier score not only lacks this property, but you don't even necessarily maintain the ordering of such scores.